3. Factorise the following using appropriate identities :
(i) 9x2 + 24xy + 16y
Answers
Answer:
done this is your answer
Answer:
Step-by-step explanation:
(
3
x
+
4
y
)
(
3
x
+
4
y
)
Explanation:
We are given:
9
x
2
+
24
x
y
+
16
y
2
We want to obtain an expression of the form:
(
a
x
+
b
y
)
(
c
x
+
d
y
)
where
a
,
b
,
c
,
d
are integers (not necessarily unique from each other).
Expanding this form we get:
a
c
x
2
+
a
d
x
y
+
b
c
x
y
+
b
d
y
2
a
c
x
2
+
(
a
b
+
c
d
)
x
y
+
b
d
y
2
From the expression we are given we must satisfy the following equations:
a
c
=
9
b
d
=
16
a
b
+
c
d
=
24
For
(
a
,
c
)
we can have (eliminating any repeats):
(
1
,
9
)
,
(
3
,
3
)
,
(
9
,
1
)
For
(
b
,
d
)
we can have (eliminating any repeats):
(
1
,
16
)
,
(
2
,
8
)
,
(
4
,
4
)
,
(
8
,
2
)
,
(
16
,
1
)
With these options, we now need to find the ones that when combined will give us the
a
b
+
c
d
=
24
:
(
1
,
9
)
(
1
,
16
)
→
1
×
1
+
9
×
16
=
145
(
1
,
9
)
(
2
,
8
)
→
1
×
2
+
9
×
8
=
74
(
1
,
9
)
(
4
,
4
)
→
1
×
4
+
9
×
4
=
40
(
3
,
3
)
(
1
,
16
)
→
3
×
1
+
3
×
16
=
51
(
3
,
3
)
(
2
,
8
)
→
3
×
2
+
3
×
8
=
30
(
3
,
3
)
(
4
,
4
)
→
3
×
4
+
3
×
4
=
24
Hence, we must choose
(
a
,
c
)
=
(
3
,
3
)
and
(
b
,
d
)
=
(
4
,
4
)
So the factored expression
(
a
x
+
b
y
)
(
c
x
+
d
y
)
is:
9
x
2
+
24
x
y
+
16
y
2
=
(
3
x
+
4
y
)
(
3
x
+
4
y
)